You've awakened my inner bird nerd! I took an ornithology course in my undergraduate studies about two years ago, and I loved it--I still geek out when an unusual bird comes into my yard! Do you happen to have any pictures of the birds that were around your campsite? Unfortunately, so little other information does make it hard to pinpoint, but at least knowing the region, time of year, and the fact that the birds in the audio are singing narrow things down a tiny bit. The book I'm using as a reference breaks down birds by family, and the descriptions include the species' voice (calls and songs; for instance, that of an American Robin is described as a sing-song "cheer-up cheerily, cheerio!")
If you have any more to go by, that would be awesome...
I'll do some more research and keep you posted!